Algorithm for the numerical calculation of the serial components of the normal form of depolarizing Mueller matrices.

The normal form of a depolarizing Mueller matrix constitutes an important tool for the phenomenological interpretation of experimental polarimetric data. Due to its structure as a serial combination of three Mueller matrices, namely a canonical depolarizing Mueller matrix sandwiched between two pure (nondepolarizing) Mueller matrices, it overcomes the necessity of making a priori choices on the order of the polarimetric components, as this occurs in other serial decompositions. Because Mueller polarimetry addresses more and more applications in a wide range of areas in science, engineering, medicine, etc., the normal form decomposition has an enormous potential for the analysis of experimentally determined Mueller matrices. However, its systematic use has been limited to some extent because of the lack of numerical procedure for the calculation of each polarimetric component, in particular in the case of Type II Mueller matrices. In this work, an efficient algorithm applicable to the decomposition of both Type II and Type I Mueller matrices is presented.

Structural circular birefringence and dichroism quantified by differential decomposition of spectroscopic transmission Mueller matrices from Cetonia aurata.

Transmission Mueller-matrix spectroscopic ellipsometry is applied to the cuticle of the beetle Cetonia aurata in the spectral range 300-1000 nm. The cuticle is optically reciprocal and exhibits circular Bragg filter features for green light. By using differential decomposition of the Mueller matrix, the circular and linear birefringence as well as dichroism of the beetle cuticle are quantified. A maximum value of structural optical activity of 560°/mm is found.

Sum decomposition of Mueller-matrix images and spectra of beetle cuticles.

Spectral Mueller matrices measured at multiple angles of incidence as well as Mueller matrix images are recorded on the exoskeletons (cuticles) of the scarab beetles Cetonia aurata and Chrysina argenteola. Cetonia aurata is green whereas Chrysina argenteola is gold-colored. When illuminated with natural (unpolarized) light, both species reflect left-handed and near-circularly polarized light originating from helicoidal structures in their cuticles. These structures are referred to as circular Bragg reflectors. For both species the Mueller matrices are found to be nondiagonal depolarizers. The matrices are Cloude decomposed to a sum of non-depolarizing matrices and it is found that the cuticle optical response, in a first approximation can be described as a sum of Mueller matrices from an ideal mirror and an ideal circular polarizer with relative weights determined by the eigenvalues of the covariance matrices of the measured Mueller matrices. The spectral and image decompositions are consistent with each other. A regression-based decomposition of the spectral and image Mueller matrices is also presented whereby the basic optical components are assumed to be a mirror and a circular polarizer as suggested by the Cloude decomposition. The advantage with a regression decomposition compared to a Cloude decomposition is its better stability as the matrices in the decomposition are determined a priori. The origin of the depolarizing features are discussed but from present data it is not possible to conclude whether the two major components, the mirror and the circular polarizer are laterally separated in domains in the cuticle or if the depolarization originates from the intrinsic properties of the helicoidal structure.

Retardation assisted enhanced Raman scattering from silicon nanostripes in the visible range.

Patterned silicon on insulator structures representing evenly spaced parallel 15 nm-thick nanostripes exhibit an enhanced Raman scattering response when excited in the visible range in an oblique incidence backscattering configuration. The enhancement phenomenon in two structures having different stripe widths, 200 and 50 nm, is investigated at various sample azimuthal orientations, excitation radiation polarizations as well as laser wavelengths and is shown to be of resonant nature. The enhanced Raman response of the patterned structures is attributed to the presence of Mie resonances, essentially resulting in the enhancement of the internal electric field within the nanostripes. It is quantitatively described in terms of the spheroid particle model extended beyond the electrostatic limit to include field retardation effects that are shown to be responsible for the resonant behaviour in the visible range.

Retrieval of a non-depolarizing component of experimentally determined depolarizing Mueller matrices.

The measurement of the Mueller matrix when the probing beam is placed on the boundary between two (or more) regions of the sample with different optical properties may lead to a depolarization in the Mueller matrix. The depolarization is due to the incoherent superposition of the optical responses of different sample regions in the probe beam. Despite of the depolarization, the measured Mueller matrix has information enough to subtract a Mueller matrix corresponding to one of the regions of sample provided that this subtracted matrix is non-depolarizing. For clarity, we will call these non-depolarizing Mueller matrices of one individual region of the sample simply as the non-depolarizing components. In the framework of the theory of Mueller matrix algebra, we have implemented a procedure allowing the retrieval of a non-depolarizing component from a depolarizing Mueller matrix constituted by the sum of several non-depolarizing components. In order to apply the procedure, the Mueller matrices of the rest of the non-depolarizing components have to be known. Here we present a numerical and algebraic approaches to implement the subtraction method. To illustrate our method as well as the performance of the two approaches, we present two practical examples. In both cases we have measured depolarizing Mueller matrices by positioning an illumination beam on the boundary between two and three different regions of a sample, respectively. The goal was to retrieve the non-depolarizing Mueller matrix of one of those regions from the measured depolarizing Mueller matrix. In order to evaluate the performance of the method we compared the subtracted matrix with the Mueller matrix of the selected region measured separately.

Null ellipsometer with phase modulation.

A new null ellipsometer is described that uses photoelastic modulator (PEM). The phase modulation adds a good signal-to-noise ratio, high sensitivity, and linearity near null positions to the traditional high-precision nulling system. The ellipsometric angles Delta and psi are obtained by azimuth measurement of the analyzer and the polarizer-PEM system, for which the first and second harmonics of modulator frequency cross the zeros. We show that the null system is insensitive to ellipsometer misadjustment and component imperfections and modulator calibration is not needed. In addition, a fast ellipsometer mode for fine changes measurement of ellipsometric angles is proposed.

Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate.

An anisotropic incoherent reflection model for the Mueller matrix elements of an optically thick uniaxial anisotropic semitransparent substrate with its anisotropy axis along its surface normal is developed. The Mueller matrix elements are measured by phase-modulated spectroscopic ellipsometry (SE) and compared with incoherent reflection model simulations. In the case of a sapphire substrate the oscillations observed are accurately modeled, and, in addition, the oscillating degree of polarization is correctly predicted. A straightforward generalization of the optical model, in the case of an arbitrary stack of layers containing a thick anisotropic semitransparent substrate, is also proposed and experimentally validated. The model is further applied to study the anisotropic dielectric function of a semi-insulating 4H-SiC wafer. An approximation based on a simple variation in the optical transition element is proposed to model the SiC birefringence. In conclusion, SE is shown to be a powerful alternative for investigating and predicting the behavior of optically thick birefringent materials.